Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

Sequences and series are often considered as a main part of part of calculus, in addition to limits, continuity, differentiation and integration.

A sequence is an enumerated collection of objects in which repetitions are allowed. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous; harmonic progressions, which is formed by taking the reciprocal of each term of an arithmetic progression; logarithmic progression, which is formed from a series whose progression getting smaller; and geometric progressions, where the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula (e.g. $a_n = 2^n + 3$), or by a recurrence relation. In a recurrence relation, the relation between the next term and the earlier terms is given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n, n \geq 0.$ Together with the initial terms $F_0=0$ and $F_1=1$, this recurrence relation defines the famous Fibonacci sequence.

A series is formed by summing a sequence. A typical question is: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is it convergent? Several tests, such as the ratio test, the root test, the limit comparison test, the integral test, etc. can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting these questions at Puzzling Stack Exchange instead.

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Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$

Everyone knows about the classic $$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$ However, is there any way to find $$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + \dfrac{1}{256} + \cdots $$
Akshaj Kadaveru
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Showing a numerical sequence converges

How could I show that the following sequence converges? $$\sum_{n = 1}^{\infty} \frac{\sqrt{n} \log n}{n^2 + 3n + 1}$$ I tried the ratio and nth-root tests and both were inconclusive. I was thinking there might be a way to use the limit comparison…
kec2013
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Why are these equations equal?

I have racked my brain to death trying to understand how these two equations are equal: $$\frac{1}{1-q} = 1 + q + q^2 + q^3 + \cdots$$ as found in http://www.math.dartmouth.edu/archive/m68f03/public_html/partitions.pdf From what I understand if I…
Parad0x13
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How find this seqence of $(n-1)a_{n}<1$

let sequence $\{a_{n}\}$ are positive numbers,and such $$(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1},\forall k\in N^{+},k\ge 2$$ show that: $(n-1)a_{n}<1,n\ge 2$ This problem is my frend ask me,and I think use introduction it Maybe this problem…
math110
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Prove that $1+{1\over1}\left(1+{1\over2}\left(1+{1\over3}\left(1+{1\over4}\left(1+{1\over5}\left( ... \right)\right)\right)\right)\right)=e$

I've found this formula, but I don't know how prove it? What's your idea for proof? $$1+{1\over1}\left(1+{1\over2}\left(1+{1\over3}\left(1+{1\over4}\left(1+{1\over5}\left( ... \right)\right)\right)\right)\right)=e.$$
user97619
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Does solution of $ x=\sum_{n=0}^\infty e^{-A_n/x}$ exist?

Usually when working with indefinite sums, I want to work out the sum or whether it convergence. But now I encountered a problem they other way around and I'm clueless... Is there even a general solution of $$ x=\sum_{n=0}^\infty e^{-A_n/x}$$ for…
JBSnorro
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Is there any connection between Legendre relation and Ramanujan's formula?

Background Consider the Legendre duplication formula: $$ \begin{aligned} \prod_{i=1}^{k-1}Γ\left(n+\frac{i}{k}\right)=(2π)^{\frac{k-1}{2}}k^{\frac{1}{2}-nk}Γ(nk) \end{aligned} $$ I chose different k values to calculate the Γ product table, but I…
Aster
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Find $\,\,e^{-1/4}\,$ to within $0.0002$

I am using Taylor's Inequality to solve this problem but this formula (page 607) is getting on my nerves and the textbook does not do a good job of explaining how to do this problem and I've gone to khanacademy and patrickjmt and they do not help…
Myles
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Use the binomial series of $(1-2x)^8$ to evaluate $0.98^8$ to 7 decimal places.

Use the binomial series of $(1-2x)^8$ to evaluate $0.98^8$ to 7 decimal places. I tried using the first five terms of the series: $1, 8, 28, 56$ and $70$, to get…
Gez Bishop
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How to compute $\sum_{k=1}^\infty \frac {1}{((2k-1)(2k+1))^2}$

I was computing a series and solved until $$\sum_{k=1}^\infty \frac {1}{((2k-1)(2k+1))^2}$$ I know Telescopic series but this doesn’t seem like it can be transform into that. Upon solving this I got $$\frac{1}{4} \sum_{k=1}^\infty \frac…
user1150809
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sum of 100 terms of logarithmic expression

Calculate value of $\displaystyle \sum^{100}_{k=1}\ln\bigg(\frac{(2k+1)^4+\frac{1}{4}}{16k^4+\frac{1}{4}}\bigg)$ My try :: $\displaystyle x^4+4y^4$ $=(x^2+2xy+2y^2)(x^2-2xy+2y^2)$ So sum $\displaystyle…
jacky
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Generalized Alternating harmonic sum $\sum_{n\geq 1}\frac{\left(1-\frac{1}{2}+\frac{1}{3}-\cdots \pm \frac{1}{n}\right)}{n^p}$

Is there a general formula for the following $$\sum_{n\geq 1}\frac{\left(1-\frac{1}{2}+\frac{1}{3}-\cdots \pm \frac{1}{n}\right)}{n^p}\,\, p\geq 1$$ What about some restrictions on $p$ , like integers or anything helpful ?
Zaid Alyafeai
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Product of $n$ power series

I have $n$ power series. How can I find the power series of the product of these $n$ series? If there are two series $(a_m)$ and $(b_m)$ then the product series $(c_m)$ is given by the Cauchy product, $$c_m = \sum_{k=0}^m a_k b_{m-k}$$ How does this…
Ramesh
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How to prove that $a_n = \frac{2n+1}{n+1}$ is bounded?

How do I prove that $\displaystyle a_n = \frac{2n+1}{n+1}$ is bounded? According to the key it works as follows. However, I don't really understand the meaning. Upper bound: $\displaystyle a_n = \frac{2n+1}{n+1} < \frac{2n+2}{n+1} =…
Fabianius
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$(n^1+n)/2$ sequence

This is going to be hard to explain so I'll just give an example Let's say we have a standard arithmetic sequence that goes up by 1 each time 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 The last number (in this case, 10) is n Scenario z: Pick 2 numbers in the…
fdj13
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